====== Estrous 1 Gender (Redux) ======

\( \require{newcommand} \)
\( \newcommand{\diff}[2]{\frac{d{#1}}{d{#2}}} \)
\( \newcommand{\abs}[1]{\left|#1\right|} \)

A second try at a QSSA analysis of the Estrous 1 Gender case.  The goal is to demonstrate how a sexual process can lead to logistic dynamics and how it deviates therefrom.

Estrous without genders.  Here we have anestrous (A) and estrous (E) individuals. N represents either type (A or E).

===== Processes =====

\[
  A \xrightarrow{\alpha} E
\]
\[
  E \xrightarrow{\gamma} A
\]
\[
  N + E \xrightarrow{\beta} N + 2 A
\]
\[
  2 N \xrightarrow{\delta} N
\]

The first two are strictly physiological processes (transitions between anestrous and estrous physiological states) while the last two are ecological (affecting the population density).

===== Full Dynamics =====

\[
  \diff{a}{t} = - \alpha a + 2 \beta n e + \gamma e - \delta a n
\]
\[
  \diff{e}{t} = + \alpha a - \beta n e - \gamma e - \delta e n
\]
\[
  \diff{n}{t} = + \beta n e - \delta n^2
\]

where \(n=a+e\).

===== Slow Timescale =====

==== Units ====

Let's assume that ecological processes are much slower than physiological ones.  In particular, let's assume

\[
  \beta \abs{K}, \delta \abs{K} \ll \alpha, \gamma
\]

where \( \abs{K} \) is a population density constant, ie. it has the same units as \(n\), \( [K] = [n] \) (where \( [\cdot] \) means "units of").  We choose \( \abs{K} \) so it is a characteristic population size for the dynamics.  That way we can compare the rate constants, because \( K \) matches the units: 

\[
  [\beta K] = [\delta K] = [\alpha] = [\gamma].
\]

==== Separation of Timescales ====

We can now specify what we mean when we say "ecological processes are much slower than physiological ones".  Let us define \( \hat{\beta} \) and \( \hat{\delta} \) so that

\[
  \beta \equiv \epsilon \hat{\beta} / \abs{K},
\]
\[
  \delta \equiv \epsilon \hat{\delta} / \abs{K}
\]

where \(\epsilon \ll 1 \) is a dimensionless constant.  Since \(\epsilon \ll 1\) the ecological processes (involving \(\beta\) and \(\delta\)) occur much less frequently than the physiological transitions (with \(\alpha\) and \(\gamma\)).

Now we can look at the dynamics under our original (fast) time variable \(t\) or with a new, slow, time variable, \( \hat{t} \equiv \epsilon t \).  In this slower timescale we find the dynamics become

\[
  \epsilon a' \equiv \epsilon \diff{a}{\hat{t}} = - \alpha a + 2 \epsilon \hat{\beta} n e / \abs{K} + \gamma e - \epsilon \hat{\delta} a n / \abs{K}
\]
\[
  \epsilon e' \equiv \epsilon \diff{e}{\hat{t}} = + \alpha a - \epsilon \hat{\beta} n e / \abs{K} - \gamma e - \epsilon \hat{\delta} e n / \abs{K}
\]
\[
  n'          \equiv \diff{n}{\hat{t}} = + \hat{\beta} n e / \abs{K} - \hat{\delta} n^2 / \abs{K}.
\]

where the prime, \( \cdot' \), denotes the slow derivative, \( \diff{\cdot}{\hat{t}} \).

Note that \(\epsilon\) cancels out in the dynamical equation for \(n\) which means that \(n\) varies at a measurable rate on the slow timescale--\(n\) is a //slow// variable.

===== Quasi-Steady-State Assumption =====

So far our analysis has been exact.  Now let's assume that the other, //fast// variables equilibrate quickly so that \( \epsilon a' \approx 0 \approx \epsilon e' \), the quasi-steady-state assumption, QSSA.  Then we can compute the quasi-steady-state densities \(a\) and \(e\) at any time:

\[
  a \approx \frac{\gamma \abs{K} + 2 \epsilon \hat{\beta} n}{(\alpha + \gamma) \abs{K} + \epsilon (2 \hat{\beta} + \hat{\delta}) n} n$\)
\[
  e \approx \frac{\alpha \abs{K}}{(\alpha + \gamma) \abs{K} + \epsilon (\hat{\beta} + \hat{\delta}) n} n.$\)

Fortunately, this gives us a way to check the validity of our assumption: we originally required \(a+e=n\) so this should still be largely correct.  Taking a Taylor expansion around \(\epsilon=0\) gives a first-order approximation:

\[
  a+e \approx n \left[ 1 + \epsilon \frac{ \alpha (\hat{\beta}-\hat{\delta}) - \gamma \hat{\delta} }{(\alpha+\gamma)^2 \abs{K}} n + \ldots \right] \equiv n \left[ 1 \pm n/n_\epsilon + \ldots \right].
\]

This tells us the QSSA is only valid when

\[
  n \ll n_\epsilon = \frac{1}{\epsilon} \frac{(\alpha+\gamma)^2 \abs{K}}{ \abs{\alpha (\hat{\beta}-\hat{\delta}) - \gamma \hat{\delta}} } = \frac{(\alpha+\gamma)^2}{\abs{\alpha (\beta-\delta) - \gamma \delta}}.
\]

==== Alternate QSSA Condition ====

<div right tip>This is too simplistic.  There are 4 Q's: \( Q=\frac{\alpha}{\delta} \left(1, \frac{1}{y}, x, \frac{x}{y} \right) \), depending on which assumption we're considering. </div>

Originally, we assumed \( \delta \abs{K} \ll \alpha \) (among other things).  That puts a condition on \(\abs{K}\) for QSSA to be satisfied:

\[
  \abs{K} \ll Q \equiv \frac{\alpha}{\delta}.
\]

===== QSSA Birthrate =====

When the QSSA holds we can approximate the population dynamics on the slow, ecological, timescale by

\[
  n' \approx \frac{\alpha \hat{\beta}}{(\alpha + \gamma) K} \frac{n}{1 + n/n_\text{Allee}} n - \hat{\delta} n^2 / \abs{K}.
\]

where
\[
  n_\text{Allee} \equiv \frac{\alpha + \gamma}{\epsilon (\hat{\beta} + \hat{\delta})} \abs{K} = \frac{\alpha+\gamma}{\beta+\delta}.$\)

<div right round box>
{{:research:draft:2009:percapitabirthrate.png}}

Figure 1: Per-capita birthrate, \(B(n)\) for population dynamics of \(n\) given QSSA assumption.
</div>

On the fast timescale this becomes

\[
  \diff{n}{t} \approx B(n) n - \delta n^2
\]

where the per-capita birthrate, \(B(n)\), is
\[
  B(n) \equiv \frac{\alpha \beta}{\alpha + \gamma} \frac{n}{1 + n/n_\text{Allee}} = r \frac{n}{n_\text{Allee} + n}
\]
and the maximum per-capita birthrate, \(r\), is
\[
  r \equiv n_\text{Allee} \frac{\alpha \beta}{\alpha + \gamma} = \frac{\alpha \beta}{\beta+\delta}.
\]

==== Logistic domain ====

If \(n_\text{Allee} \ll n \ll n_\epsilon\) then the dynamics lie within the //logistic domain// where the per-capita birthrate is roughly constant, \(B(n)\approx r\), and the population dynamics follow

\[
  \diff{n}{t} \approx r n - \delta n^2,
\]

the logistic rate equation.

==== Allee domain ====

On the other hand, if the population is small, \( n \ll n_\text{Allee} \), then the birthrate is suppressed, the Allee effect.  In this domain the per-capita birthrate is roughly \(B(n)\approx r n / n_\text{Allee} \) and the population follows

\[
  \diff{n}{t} \approx \left( \frac{r}{n_\text{Allee}} - \delta \right) n^2.
\]

Clearly, the population is unstable if \( r < \delta n_\text{Allee} \).  This would reflect a //strong// Allee effect, where the mating rate drops to an unsupportable level if the population is too small.  On the other hand, if \( r > \delta n_\text{Allee} \) (or \( \alpha \beta > \delta (\alpha+\gamma) \)) then we have a //weak// Allee effect--the birthrate is suppressed compared to the logistic equation but the population will nevertheless grow if undisturbed.

===== Carrying capacity =====

Regardless of whether QSSA holds we can find the non-trivial equilibrium population density. Since this represents the //characteristic// population size we can choose this to represent our parameter \(K\) from above:

\[
  K \equiv \frac{r}{\delta} - n_\text{Allee} = \frac{\alpha \beta - (\alpha + \gamma) \delta }{(\beta + \delta) \delta}.
\]

(The first equality is expressed in terms of the above QSSA analysis but the second equality represents the equilibrium \(n\)-density even if QSSA is violated.)

I haven't done the Jacobian stability analysis for \(da/dt, de/dt\) but it looks pretty clear that \(n=K\) is only stable when \(K>0\), otherwise \(n=0\) is stable.  Then if \(K\leq 0\) the population is unstable and will inevitably go extinct.  So we only expect \(K>0\) to represent naturally-occurring populations (unless the parameters have changed recently to destabilize the population).

If \(K>0\) then the //weak// Allee effect applies in the Allee domain so the population is globally stable (regardless of how small it gets).  On the other hand, if \(K<0\) then the system is unstable for all \(n\).

One final point about the carrying capacity.  For comparison with the logistic equation, in terms of \(r\), \(K\), and \(n_\text{Allee}\) the QSSA dynamics can be written

\[
  \diff{n}{t} = r n \left( \frac{n}{n+n_\text{Allee}} - \frac{n}{K+n_\text{Allee}} \right).
\]

===== Parameter space =====

The possible dynamics can be expressed more concisely by reducing the parameter space to two dimensions: 

\[
  x=\gamma/\alpha
\]
\[
  y=\beta/\delta.
\]

We will find there are five qualitatively separate regions in this parameter space, depending on the relationships between the constants \(n_\epsilon, K\), \(n_\text{Allee}\), and \(Q\).  In terms of \(x\) and \(y\) these constants can be written as

\[
  n_\epsilon = \frac{\alpha}{\delta} \frac{(1+x)^2}{\abs{y-1-x}}
\]
\[
  K = \frac{\alpha}{\delta} \frac{y-1-x}{1+y}
\]
\[
  n_\text{Allee} = \frac{\alpha}{\delta} \frac{1+x}{1+y}
\]
\[
  Q = \frac{\alpha}{\delta}.
\]

==== Stability ====

If \(K\leq 0\) then we expect the population to be unstable and to crash (under both the full dynamics and the QSSA approximation).  In terms of \((x,y)\) this is

\[
  \text{Unstable} : y\leq 1+x.
\]
!! QSSA limit

If \( K > 0\)  (\(y>1+x\)) then \(K\) represents a characteristic population density (actually, the equilibrium) so it is reasonable to expect the actual density to be on the same order, \( n \sim K \).  Then we can express the QSSA condition as \( K \ll n_\epsilon \).  This condition implies

\[
  \frac{y-1-x}{1+y} \ll \frac{(1+x)^2}{y-1-x}
\]

or

\[
  \text{QSSA} : y \ll (3+x)(1+x).
\]

=== Alternative QSSA limit ===

<div right tip>This becomes more complicated because there are 4 values of Q we need to compare to K.  It turns out the QSSA regime gets very small (around \(y=1+x\)) for \(x<1\) and grows slowly (linear with \(x\) for \(x>1\).  In all, Q is a poor predictor of when QSSA applies -- \(n_\epsilon\) is better.</div>

\(\abs{K}\) represents the characteristic population density.  A QSSA condition was \( \abs{K} \ll Q = \alpha / \delta \), so in terms of \((x,y)\) this becomes

\[
  y-1-x \ll 1+y \text{ if } K\geq 0
\]
\[
  1+x-y \ll 1+y \text{ if } K<0.
\]

The first form is approximately valid for all \((x>0, y>1+x)\).  The second form, which occurs in the unstable regime, implies

\[
  \text{unstable QSSA} : y \gg \frac{x}{2}.
\]

==== Logistic vs Allee domain ====

If \(K>0\) and the QSSA applies then the dynamics will carry \(n\) towards \(K\).  If \(K<n_\text{Allee}\) then the population will always be in the Allee domain with a depressed population growth rate.  This condition is simply

\[
  \text{Allee} : y < 2(1+x).
\]

==== Summary ====

<div right round box>
{{:research:draft:2009:parameterspace.png}}

Figure 2: Parameter space can be divided into five regions characterizing the dynamics.  The red dots indicates points where the dynamics are computed numerically (see below).
</div>

In conclusion, there are five regions of interest in the \((x,y)\)-plane:

\[
  \text{stable non-QSSA} : K \gtrsim n_\epsilon > n_\text{Allee} > 0
\]
\[
  \text{stable logistic QSSA} : n_\epsilon \gg K > n_\text{Allee} > 0
\]
\[
  \text{stable Allee QSSA} : n_\epsilon \gg n_\text{Allee} > K > 0
\]
\[
  \text{unstable QSSA} : n_\epsilon \gg n_\text{Allee} > 0 > K > -Q
\]
\[
  \text{unstable non-QSSA} : n_\epsilon \gg n_\text{Allee} > 0 > -Q > K
\]

Note that \(n_\epsilon > n_\text{Allee}\) always.

===== Numerics =====

To test our analytic results we can construct a numeric simulation.  The above analysis suggests the regions we should check numerically (marked in red on Figure 2).  The results, shown below, are better than could be expected.  The QSSA tracks the full dynamics qualitatively in all cases and quantitatively for all \(x>1\).  That makes sense because the valid QSSA regions get //squished// from above and below as \(x\leftarrow 0\).  As \(x\) increases, the QSSA regions grow.

The QSSA is a more robust approximation than would be expected from the conditions \(n\ll Q\) or \(n\ll n_\epsilon\).  

|                             Region                              |                   \( x = 0 \)                    |                    \( x = 1 \)                    |                   \( x = 4 \)                    |                    \( x = 16 \)                    |
|     Stable non-QSSA \\ \( K > n_\epsilon > n_{Allee} > 0 \)     |     {{estrous20091002_stablenon_x0y3.5.png?150}} |       {{estrous20091002_stablenon_x1y10.png?150}} |    {{estrous20091002_stablenon_x4y47.5.png?150}} |    {{estrous20091002_stablenon_x16y467.5.png?150}} |
|        Logistic \\ \( n_\epsilon > K > n_{Allee} > 0 \)         |      {{estrous20091002_logistic_x0y2.5.png?150}} |         {{estrous20091002_logistic_x1y6.png?150}} |     {{estrous20091002_logistic_x4y22.5.png?150}} |     {{estrous20091002_logistic_x16y178.5.png?150}} |
|          Allee \\ \( n_\epsilon > n_{Allee} > K > 0 \)          |         {{estrous20091002_allee_x0y1.5.png?150}} |            {{estrous20091002_allee_x1y3.png?150}} |         {{estrous20091002_allee_x4y7.5.png?150}} |         {{estrous20091002_allee_x16y25.5.png?150}} |
|    Unstable QSSA\\ \( n_\epsilon > n_{Allee} > 0 > K < -Q \)    |  {{estrous20091002_unstableqssa_x0y0.5.png?150}} |  {{estrous20091002_unstableqssa_x1y1.25.png?150}} |  {{estrous20091002_unstableqssa_x4y3.5.png?150}} |  {{estrous20091002_unstableqssa_x16y12.5.png?150}} |
|  Unstable non-QSSA\\ \( n_\epsilon > n_{Allee} > 0 > -Q > K \)  |     {{estrous20091002_unstablenon_x0y0.png?150}} |   {{estrous20091002_unstablenon_x1y0.25.png?150}} |     {{estrous20091002_unstablenon_x4y1.png?150}} |      {{estrous20091002_unstablenon_x16y4.png?150}} |

